Can't speak from personal experience of dealing with parents but educate yourself with info about ethics, environment, nutrition etc so you can defend yourself from any criticisms. In particular, be aware of the studies that show advantages to veganism on cancer, longevity and heart disease.
Offer to learn to cook a few vegan meals for the family.
Tell the family this is very important for you. Suggest that they watch a movie about veganism with you to understand it.
Expect resistance/uncertainty, learn to deal with it steadily over time by working on people that are not initially accepting.
Jamie offers good advice and one of the newest and best films is "What the Health", watching that together would be good.
Also, do your research and be prepared when you speak to them. Show them what you plan to eat and why and maybe even use Crometer to show them how you would be getting all the nutrients you need. Make sure you are getting enough calories from whole foods like fruits, veggies, greens, beans, potatoes, etc. You could also show them how it will cost less than if you are eating animal products and ask them if they would consider joining you for a week or even for a weekend.
If I were you, I'd just get it out there. The fact you feel the need to hide it from them suggests to me that perhaps they're not the most open minded people. Just tell them what you're doing, why you're doing it, how you are not going to change and you've made up your mind and you hope they respect your opinion, but if they don't, they're going to have to accept it.
That's true, but no parent is going to force their child to starve and no parent only feeds their kid meat. Surely they are already purchasing vegan products like rice and veggies. If they see she refuses to eat anything but the vegan items in the household, if they care about her, they'll buy more of those so she gets enough food.